The Anderson-Higgs Mechanism

One reason for this posting is that exchanges in the comment section of the previous one led me to look into some history, and I found some odd and possibly interesting facts I hadn’t previously known. So, part of this will just be lifting of some links from comments in the last posting.

Another reason is that while the history may seem obscure, what’s at issue is the central unsolved problem of particle physics: the nature of electroweak symmetry breaking, and no excuse for thinking more about this topic should be let to pass by. The Yang and Mills work on non-abelian gauge theory published in 1954 had one huge problem: in perturbation theory it has massless particles which don’t correspond to anything we see. One way of getting rid of this problem is now fairly well-understood, the phenomenon of confinement realized in QCD, where the strong interactions get rid of the massless “gluon” states at long distances (they are relevant at short distances, visible in terms of jets seen at colliders).

By the very early sixties, people had begun to understand another source of massless particles: spontaneous symmetry breaking of a continuous symmetry. If the vacuum state is non-invariant under a continuous symmetry, you expect to find one massless state in the theory for each generator of the symmetry. These are called “Nambu-Goldstone” particles, and pions provide an example (only approximately massless, since the symmetry is approximate).

What Philip Anderson realized and worked out in the summer of 1962 was that, when you have both gauge symmetry and spontaneous symmetry breaking, the Nambu-Goldstone massless mode can combine with the massless gauge field modes to produce a physical massive vector field. This is what happens in superconductivity, a subject about which Anderson was (and is) one of the leading experts. His paper on the subject was submitted to Physical Review that November, and appeared in the April 1963 issue of the journal, in the particle physics section. It explains what is commonly called the “Higgs mechanism” in very much the same terms that the subject appears in modern particle physics textbooks and notes:

It is likely, then, considering the superconducting analog, that the way is now open for a degenerate-vacuum theory of the Nambu type without any difficulties involving either zero-mass Yang-Mills gauge bosons or zero-mass Goldstone bosons. These two types of bosons seem capable of “canceling each other out” and leaving finite mass bosons only.

All that is missing here is an explicit relativistic example to supplement the non-relativistic superconductivity one. This was provided by several authors in 1964, with Higgs giving the first explicit relativistic model. Higgs seems also to have been the first to explicitly discuss the existence in models like his of a massive mode, of the sort that we now call a “Higgs particle”, the target of active searches at the Tevatron and LHC.

So it was probably completed summer ’62. Very little attention was paid to it except that in fact— well, Higgs reinvented it. In some ways the particle physicists tell me had less understanding; in some ways he had more. He certainly made a real model out of it where I had only a mechanism…

about the Anderson-Higgs phenomenon, if I may use the word. In the paper that I wrote I definitely said people have been worried about the Goldstone boson in broken symmetry phenomena. The Goldstone boson is not necessary. Here is the possibility of removing the Goldstone boson, mixing it with a gauge boson, and ending up with zero mass. [should be “non-zero” maybe a transcription error]…So I think I really understood the nature of the mechanism…

It was not published as a paper in the Condensed Matter Physics. It was published as a paper in Particle Physics. Brout paid attention to it. And he and Englert two years later produced a model of symmetry breaking, which if you’ll read carefully the summary of their work that t’Hooft and Veltman give (Nobel Prize winner this year), they say that they took off very much from the Brout-Englert paper, and there’s no way Brout was not perfectly aware of my work and I would be surprised if the Brout Englert paper doesn’t reference it rather than Higgs or along with Higgs. So in fact it didn’t fall completely on deaf ears.

Note added 5/15/2013: I’ve heard from Martin Veltman that at the time they were working on the renormalizability of Yang-Mills, he and ’t Hooft were not aware of the Brout/Englert work, or of the general issues about the Goldstone theorem and the Higgs mechanism that Brout/Englert and others were addressing. Veltman’s Nobel lecture describes the history in detail, and has nothing like what Anderson describes (neither does ’t Hooft’s).

Given the background Brout had in condensed matter physics and Anderson’s claim that “there’s no way Brout was not perfectly aware of my work”, it is quite surprising that no reference to Anderson occurs in the paper he and Englert published in Physical Review Letters. It arrived at the journal June 26, 1964 and came out in an issue dated August 31, 1964. In historical talks about this given back in 1997 (available here), Brout and Englert write:

We knew from our study of ferromagnetism that long range forces give mass to the spin waves and we were aware, from Anderson’s analysis of superconductivity [5], of the fact that the massless mode of neutral superconductors, which is also a Nambu-Goldstone mode, disappears in charged superconductors in favor of the usual massive plasma oscillations resulting from the long range coulomb interactions in metals. Comforted by these facts, we decided to confront, in relativistic field theory, the long range forces of Yang-Mills gauge fields with the Nambu-Goldstone bosons of a broken symmetry.

The latter arose from the breaking of a global symmetry and Yang-Mills theory extends the symmetry to a local one [6]. Although the problem in this case is more subtle because of gauge invariance, the emergence of the Nambu-Goldstone massless boson is very similar. We indeed found that there were well defined gauges in which the broken symmetry induces such modes. But, as we expected, the long range forces of the Yang-Mills fields were conflicting with those of the massless Nambu Goldstone fields. The conflict is resolved by the generation of a mass reducing long range forces to short range ones. In addition, gauge invariance requires the Nambu-Goldstone mode to combine with the Yang Mills excitations. In this way, the gauge fields acquire a gauge invariant mass!

This work was finalized in 1964.

Very oddly, the only reference to Anderson’s work that they give (their [5]) is to a 1958 paper of his, not to the 1963 paper which had the same conclusions as theirs, a year earlier.

Brout and Englert don’t give a full model, just assume existence of a scalar field with spontaneously broken symmetry, and specified couplings to the gauge fields. Working independently, Peter Higgs in July 1964 sent a paper to Physics Letters arguing that, even relativistically, Anderson’s argument worked and there is no need for massless particles in the case of spontaneous symmetry breaking with a local symmetry. This paper was published, but a paper he sent a week later in which he wrote down an explicit model (the Abelian Higgs model) was rejected. It was later submitted to (August 31, 1964) and accepted at Physical Review Letters (published in the October 19, 1964 issue), where the referee (Nambu) made Higgs aware of the Brout-Englert paper, which Higgs refers to in a footnote. The Higgs paper does refer to Anderson’s 1963 paper, writing in the introduction:

This phenomenon is just the relativistic analog of the plasmon phenomenon to which Anderson [3] has drawn attention.

Higgs gives his version of the history here, and refers to the “Anderson mechanism”, writing:

During October 1964, Higgs had discussions with Gerald Guralnik, Carl Hagen and Tom Kibble, who had discovered how the mass of non-interacting vector bosons can be generated by the Anderson mechanism.

Guralnik, Hagen and Kibble had been working on what Higgs calls the “Anderson mechanism” and Anderson the “Anderson-Higgs mechanism”, writing a paper about it for submission to PRL. Guralnik gives his version of the history here (writing about the “Brout, Englert, Guralnik, Hagen, Kibble, Higgs phenomenon”, Higgs last, no Anderson), Kibble’s is here. In Guralnik’s version:

as we were literally placing the manuscript in the envelope to be sent to PRL, Kibble came into the office bearing two papers by Higgs and the one by Englert and Brout. These had just arrived in the then very slow and unreliable (because of strikes and the peculiarities of Imperial College) mail. We were very surprised and even amazed. We had no idea that there was any competing interest in the problem, particularly outside of the United States. Hagen and I quickly glanced at these papers and thought that, while they aimed at the same point, they did not form a serious challenge to our work.

His explanation for why they did not refer to Anderson is:

At the same time, Kibble brought our attention to a paper by P.W. Anderson [26]. This paper points out that the theory of plasma oscillations is related to Schwinger’s analysis of the possibility of having relativistic gauge invariant theories without massless vector particles. It suggests the possibility that the Goldstone theorem could be negated through this mechanism and goes on to discuss “degenerate vacuum types of theories” as a way to give gauge fields mass and the necessity of demonstrating that the “necessary conservation laws can be maintained.” In general these comments are correct. However, as they stand, they are entirely without the analysis and verification needed to give them any credibility. These statements certainly did not show the calculational path to realize our theory and hence the unified electroweak theory. It certainly did not even suggest the existence of the boson now being searched for at Fermi lab and LHC. The actual verification that the same mechanism actually worked in non-relativistic condensed-matter theories as in relativistic QFT had to wait for the work of Lange [28], which was based on GHK. We did not change our paper to reference the Anderson work.

See Guralnik’s paper for a detailed discussion of those points which he feels Anderson, Brout, Englert and Higgs had missed about all this. It remains true that the full understanding of how this works non-perturbatively is rather tricky, especially in the chiral, non-perturbative context that is relevant to the Standard Model. It may very well be that there is some important piece of understanding about this that has been missing and will someday lead to a final understanding of the origin of electroweak symmetry breaking.

Update: For two other recent expository articles about this subject and its history, see here and here.

What about Ernst Stueckelberg? Wikipedia says “In 1938 he recognized that massive electrodynamics contains a hidden scalar, and formulated an affine version of what would become known as the Abelian Higgs mechanism.”

In the Discussion section somebody writes: “his despite the fact that he invented the renormalization group, despite the intermediate bosons, despite covariant perturbation theory, and despite the first Abelian Higgs mechanism, in 1957, remember this is same year as BCS, before Anderson, before Brout. His lack of recognition is a notable and sad fact.”

Does anyone know more about this? I’m always amazed “Stueckelberg” pops up so many times in priority discussions. As far as I know, Feynman is the only Famous Physicist that acknowledges his work.

My information comes from “The Second Creation” by Crease and Mann. Ernst Stueckelberg was at the University of Zurich. Wolfgang Pauli was the big cheese at nearby ETH Zurich. From what I can tell, Stueckelberg suffered from low self-esteem. Pauli was a disaster for Stueckelberg. Pauli mercilessly put down Stueckelberg’s ideas. Stueckelberg published his papers in French, and in obscure journals. It is also recognized that Stueckelberg’s papers were very difficult to read. It is only with hindsight that we recognize much of Stueckelberg’s contributions.

On the subject of Yang-Mills and massive gauge vector bosons, it is a matter of record that Pauli formulated models of field theories with non-Abelian local gauge invariance before Yang and Mills. But Pauli also realized that the inclusion of a nonzero bare mass term for the gauge bosons violated gauge invariance and made the theory unrenormalizable. And on this basis Pauli rejected locally non-Abelian gauge theories as valid candidates for physics. It is a matter of record that Pauli attacked C.N. Yang on this basis. Read articles by Jeremy Bernstein on the subject.

Pauli is perhaps admired for the phrase “Not Even Wrong” but Pauli also put down much that was Not Even Right.

What Stueckelberg did uses a scalar field to make a gauge field massive in a different way than the Anderson-Higgs mechanism. It’s generally called the “Stueckelberg mechanism”, and doesn’t seem to be relevant to the electroweak symmetry breaking of the standard model. For more about it, see here

Some other interesting notes in this plot that you may have (or not) uncovered.

The 1979 winners (AS and SW) primarily worked from the GHK paper due to the completeness and proximity to each other at Imperial College London. All six Sakurai winners are referenced in the 1979 Nobel talks by AS and SW. AS mentions that Kibble “tutored” him on these concepts.

It is odd that the Brout-Englert Historical Account posted in blog does not mention GHK – why introduce anything to the history that ruins “Nobel Math” of three. So this is in-line with how they treated PA.

t’Hooft and Veltman’s story has changed over time. They referenced all three 1964 PRL papers and actually refer to it as the “Higgs-Kibble Mechanism” on page one of their Nobel Paper (see link below). Since being buttressed by their Nobel they have pushed a BEH award – particularly BE.

Peter, if you were a member of the Nobel Prize comitee after the discovery of the Higgs boson at the LHC, what would be your decision ?
Who deserve the prize ? And don’t forget the rule : no more than 3 names

In its own way the 2010 Sakurai Prize reflects the prejudices of HEP. Even though Anderson’s 1963 paper was published in the Particle Physics section of the Physical Review, it is nevertheless viewed as condensed matter (and the work is of course nonrelativistic).

Just for the reason of completeness I would like to mention
the paper by A.A. Migdal and A.M. Polyakov in
Sov.J. -JETP 24 , 91 (1966)
“Spontaneous Breakdown of Strong Interaction Symmetry and the Absence of Massless Particles”

In some sense the right Nobel for that has already been given (Weinberg-Salam), but if one wanted to go further back, I think Anderson + Higgs would be justifiable, Anderson for discovering the Higgs mechanism, Higgs for first writing down a relativistic theory implementing the Anderson mechanism and noticing that it had a physical mode that would be the Higgs particle.

But, in any case I’m hoping that what the LHC discovers involves some much more interesting mechanism for electroweak symmetry breaking than a Higgs field.

nice post peter–’tis why this is one of the best science blogs out there.

speaking of blogs, anderson should have blogged his work in 1962!

all kidding aside, i imagine the internet, and the numerous forums/blogs/archives will help us prevent such controversy in the future.

the knowledge that one cannot easily get away with forgetting to reference work will work wonders, as well as the permanent record via blogs/arxiv/groups/forums/facebook/blogs, all of which are dated and the vast majroity of which can be trusted, as it would be impossible to hack a facebook date!

thoughts? comments?

also, it appears journals are becoming less relevant, with their long delays vs. the immediacy of the internet.

FWIW there’s a whole book on this stuff (reprints of papers). The papers by Anderson (1958, 1963), Nambu, Schwinger (2D electrodynamics) are all there. Migdal-Polyakov also. But the papers by Higgs, Brout-Englert and Guralnik-Hagen-Kibble are not reprinted. I suppose the book is about “dynamical” symmetry breaking ~ BCS superconductivity? ~ although in the HEP context I have no idea what would be a condensate bilinear in fermion fields.

Comment 1
I believe that the first person who pointed out that a gauge boson can acquire a nonzero physical mass in the relativistic quantum field theory is J. Schwinger. He solved 2-dimensional massless QED, called the Schwinger model; the gauge field becomes massive (PR 128(1962),2425). At that time, this was called Schwinger mechanism, but it turns out to be nothing but the dynamical Higgs mechanism.

Comment 2
Higgs did not resolve the dilemma between the Goldstone theorem and the Higgs mechanism. He merely avioded the
contradiction by adopting Coulomb gauge. The Goldston theorem holds in the manifesly covariant field theory, and the requirement of manifest covariance is very important backbone of the Standard Theory. I emphasize that the Nambu-Goldstone boson does exist in the electroweak theory.
It is merely unobservable by the subsidery condition (Gupta condition). Indeed, without NG boson, the charged pion could not decay into muon and antineutrino (or antimuon and neutrino) because the decay through W-boson violates angular-momentum conservation.

Comment 3
I know that it is a common belief that pion is regarded
as an “approximate” NG boson. But it is quite strange to regard pion as an almost massless particle. It is equivalent to regard nuclear force as an almost long-range force! The chiral invariance is broken in the electroweak theory. And as I stated above, the massless NG boson does exist.

I would agree about Schwinger and his conclusions based on his model. GHK stated that in the Lorentz gauge a NG boson would exist but it would be “pure gauge” and not observable. (Higgs did not say this, in his 1964 paper anyway.)

The decay of a charged pion principally into (muon, mu_antineutrino) and not (electron, e_antineutrino) has long been regarded as a demonstration of the V-A structure of the weak interactions. The resulting helicity suppression is the reason that the branching ratio for the decay to (electron, e_antineutrino) is so small, despite having a much larger phase-space. “Indeed, without NG boson, the charged pion could not decay into muon and antineutrino … the decay through W-boson violates angular-momentum conservation.” ??

Explanation on Comment 2
Pion’s spin is zero, while W-boson’s spin is one. People usually understand that the pion decays into a muon and a neutrino through an intermediate state consisting of one W-boson. But this is forbidden by the angular-momentum conservation law in the rest frame of the pion. Note that conservation law (
possibly except for energy because it is canonically conjugate to time) must hold even in intermediate states.

“… scrabbling for the exit …” There’s probably a Monty Python sketch for that. More likely there will be a generous attribution of credit to the *other* person. For example the Higgs may be renamed the Kibble, or the Brenglert, or all the HEP theorists will join forces to point the finger at Anderson. But it will he hard to conclude that the Higgs does not exist simply because it is not found. The mass limit (with 95% confidence) will be pushed up. Eventually the limit may reach outrageous levels, but that may require more than the LHC.

Regarding Schwinger, there is one big problem in the argument that he discovered the dynamical symmetry breaking mechanism we usually call Higgs mechanism: in 2D there is no spontaneous symmetry breaking (Mermin-Wagner theorem). His paper is a well known landmark and recognized for its contribution towards understanding the confinement phenomenon and effective field theories (bosonization). On the other hand, it is also well known, that it misses one of the most prominent features of 2D QED (and 4D QCD), the theta-vacuum structure, entirely.

The Schwinger model is an early toy model study of QCD. I guess that therefore – by a very large leap of faith – one might even call it a grandfather in spirit to technicolor theories and therefore an early precursor study of a dynamical Higgs phenomenon. (This notwithstanding that the fact that the condensate one would like to associate with the Higgs is simply not formed in 2D because that would violate the Mermin-Wagner theorem) . But attributing to it the discovery of the Higgs mechanism is way beyond reasonable argument.

In all this discussion about the early years of the Higgs et alii “mechanism” we should not forget that another link was searched for, once in a while, by considering a scale covariant scalar field as a bridge to gravity, e.g.:
F. Englert, E. Gunzig, C. Truffin, P. Windey 1975 (Conformal invariant general relativity with dynamical symmetry breakdown,Phys. L. 57B, 73ff.) tried a link to Jordan-Brans-Dicke theory.
L. Smolin 1979 (Towards a theory of spacetime structure at very short distances, Nucl. Phys. B 160, 253ff.) started to play around with Weyl geometric gravity of the Dirac-Utiyama type.

Every ten to fifteen years similar ideas seemed to have popped up, but remained marginal annotations to the main discourse. If something came out of these attempts, would n’t that be already a bit “more interesting” than the ordinary Higgs field; or do you expect even more, Peter?

Your assertion that in 2D there is no spontaneous symmetry breaking is not valid in the indefinite-metric quantum field theory. The specialty of 2D is the non-existence of the 2-point funtion of a massless scalar field in the Hilbert space, but it does exist in the indefinite-metric space.

The theta vacuum business appears in the massive Schwinger model but not in the massless Schwinger model. Furthermore, the appearance of the theta vacua depends on the choice of the representation; it is possible to avoid the theta vacua.

I of course do not regard Schwinger as the discoverer of the Higgs mechanism. He is a precursor. Probably, Anderson should also be regarded as a precursor.

The manifestly covariant formulation of the Schwinger model contains massless ghost particles. Gauge symmetry is broken so as to give a mass to the gauge particle. The ghost particle, which is unphysical because of the subsidiary condition, plays the role of the Goldstone boson.

I am sorry, but there is no dynamical symmetry breaking in the Schwinger model. In your work, you explicitly broke the gauge symmetry by a gauge fixing condition. Of course ghosts will arise then – but they are unphysical.

The mass of the Schwinger boson is due to the anomaly term. This is not symmetry breaking, there was no symmetry to begin with.

You seem not to understand the fundamental points of the quantum theory of a gauge field. Let me explain. Any classical theory which has local symmetry cannot be quantized without violating it by introducing gauge fixing. Instead, the theory should become BRS invariant; the BRS symmetry is the quantum version of local gauge symmetry. For the abelian gauge theory, FP ghosts decouple, so only the B field remains.
Although local symmetry does not exist in quantum gauge theory, global symmetry remains unbroken if spontaneous symmetry breaking does not occur. I emphasize that local gauge symmetry plays no role in the Higgs mechanism of the quantum gauge theory. Local symmetry is a classical concept!

The Schwinger mechanism is NOT an anomaly. It is not good to regard anything which you cannot understand as anomaly.
The Schwinger model has essentially the same mechanism as that of the chiral gauge theory.

Peter Higgs (PRL, October 1964)
1 – Yes
2 – Yes
3 – No. In his Physics Letters Paper PH stated Goldstone theorem could fail in radiation but does not show how. In PRL paper, PH does not pick a gauge nor did he not give a quantum mechanical argument to the Goldstone theorem which is purely a quantum phenomenon.
4 – Yes

The papers have gone down in history together as they should. It is a shame the Nobel “rule of three” tries to segment these as TOGETHER they each brought different pieces to this and formed the basis of the standard model. Again, nice post Peter. Certainly Anderson’s work was also critical to this whole phenomena and period.